Theorem summodclem2 12076

Description: Lemma for summodc 12077. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
summodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5672	. . . . 5 ⊢ (𝑚 = 𝑎 → (ℤ≥‘𝑚) = (ℤ≥‘𝑎))
2	1	sseq2d 3270	. . . 4 ⊢ (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑎)))
3	1	raleqdv 2749	. . . 4 ⊢ (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10819	. . . . 5 ⊢ (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
5	4	breq1d 4121	. . . 4 ⊢ (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
6	2, 3, 5	3anbi123d 1349	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
7	6	cbvrexv 2781	. 2 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8		simplr3 1068	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9		simplr1 1066	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑎))
10		uzssz 9880	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑎) ⊆ ℤ
11	9, 10	sstrdi 3252	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
12		1zzd 9609	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
13		simprl 531	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
14	13	nnzd 9705	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
15	12, 14	fzfigd 10800	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
16		simprr 533	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
17		f1oeng 6998	. . . . . . . . . . . . . 14 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
18	15, 16, 17	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
19	18	ensymd 7025	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
20		enfii 7131	. . . . . . . . . . . 12 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
21	15, 19, 20	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
22		zfz1iso 11221	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
23	11, 21, 22	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24		isummo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
25		simplll 535	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26		isummo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
27	25, 26	sylan 283	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
28		eleq1w 2295	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
29	28	dcbid 846	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
30		simpr2 1031	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
31	30	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
32		simpr 110	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → 𝑘 ∈ (ℤ≥‘𝑎))
33	29, 31, 32	rspcdva 2928	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → DECID 𝑘 ∈ 𝐴)
34		summodclem2.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
35		eqid 2234	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0))
36		simprll 539	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37		simpllr 536	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38		simplr1 1066	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑎))
39		simprlr 540	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
40		simprr 533	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 12075	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
42	41	expr 375	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
43	42	exlimdv 1868	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
44	23, 43	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45		climuni 11986	. . . . . . . . 9 ⊢ ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
46	8, 44, 45	syl2anc 411	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
47	46	anassrs 400	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48		eqeq2 2244	. . . . . . 7 ⊢ (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦 ↔ 𝑥 = (seq1( + , 𝐺)‘𝑚)))
49	47, 48	syl5ibrcom 157	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
50	49	expimpd 363	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
51	50	exlimdv 1868	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
52	51	rexlimdva 2662	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
53	52	r19.29an 2687	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
54	7, 53	sylan2b 287	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ⦋csb 3140   ⊆ wss 3213  ifcif 3622   class class class wbr 4111   ↦ cmpt 4173  –1-1-onto→wf1o 5353  ‘cfv 5354   Isom wiso 5355  (class class class)co 6052   ≈ cen 6975  Fincfn 6977  ℂcc 8130  0cc0 8132  1c1 8133   + caddc 8135   < clt 8313   ≤ cle 8314  ℕcn 9242  ℤcz 9582  ℤ_≥cuz 9859  ...cfz 10348  seqcseq 10816  ♯chash 11146   ⇝ cli 11971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-seqfrec 10817  df-exp 10908  df-ihash 11147  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972

This theorem is referenced by:  summodc  12077